Since its inception over fifty years ago, the field of artificial intelligence has remained separated from the field of psychology by the fundamental mathematical dichotomy between the discrete and the continuous. While computer scientists regard symbols as discrete bit patterns, psychologists regard them as continuous visual icons. The substitution of integral signs for summation signs sometimes serves to create a bridge across this dichotomy, for example, between the continuous and discrete versions of the Fourier transform.
It is easier to design a symbol processing system on a digital computer by representing symbols as discrete bit patterns, than to design a system that represents symbols as cumbersome digitized images. In biological systems, the central nervous system (CNS) represents and operates on symbols as continuous images because it was forced to do so. Throughout the course of evolution, everything in nature, predator and prey, the external environment and the body itself, was governed by the laws of physics, which are based on functions defined in space and time. In the struggle to survive, an organism had to process, as quickly as possible, continuous symbol representations.
The lack of a complete and thorough scientific understanding of animal intelligence and human consciousness has limited the ability to build computer systems that mimic these capabilities. The present invention provides a new model that may provide insight into how to design such systems. The design connects together a massive number of relatively small processing elements to form a large recurrent image association system. It is completely parallelizable, and all of the computations take place recursively over small, localized regions.
The mathematical definition of a manifold is a metric space, which is everywhere locally homeomorphic to an open set in Rn, where R denotes the real numbers (Spivak, 1979; Schutz, 1980). Less formally, we can regard a one-dimensional manifold as a smooth curve, a two-dimensional manifold as a smooth surface, akin to a deformed balloon, and analogously for higher order manifolds. In order to accommodate discontinuities in the real world, we use Borel measurable functions to represent data, and define operations using Lebesgue integrals (Kolmogorov & Fomin, 1970; Royden, 1988). For brevity, in the following discussion, we will refer to generalized or Borel measurable functions simply as functions.
Since open regions of space, time and frequency as well as their product spaces are all manifolds, one can accurately describe virtually everything in nature, as a function defined on a manifold. At the level of quantum physics, matter and energy are discrete. Nevertheless, as is the case for the electrons that define the current density J in Maxwell's equations, and the molecules that define the density of matter ρ in the Navier-Stokes equations, at the macroscopic level these quantities are differentiable.